The appearance of non trivial torsion for some Ricci dependent theories in the Palatini formalism

Abstract

As is known from studies of gravity models in the Palatini formalism, there exist two inequivalent definitions of the generalized Ricci tensor in terms of the generalized curvature namely, $\widetilde{R}_{μν}=R^ρ_{μρν}$ and $R^μ_ν=g^{αβ}R^μ_{ανβ}$. A deep formal investigation of theories with lagrangians of the form $L=L(\widetilde{R}_{(μν)})$ was initiated in [4]. In that work, the authors leave the connection free, and find out that the torsion only appears as a projective mode. This agrees with the widely employed condition of vanishing torsion in these theories as a simple gauge choice. In the present work the complementary scenario is studied namely, the one described by a lagrangian that depends on the other possible Ricci tensor $L=L(R_{(μν)})$. The torsion is completely characterized in terms of the metric and the connection, and a rather detailed description of the equations of motion is presented. It is shown that these theories are non trivial even for $1+1$ space time dimensions, and admit non zero torsion even in this apparently simple case. It is suggested that to impose zero torsion by force may result into an incompatible system. In other words, the presence of torsion may be beneficial for insuring that the equations of motion are well posed. The results presented here do not contradict the results of [4], as the resulting theories are inequivalent.